Continuing my Math Education - Book Recommendations Please

In My Humble Opinion
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Here’s the scoop. I have a degree in Math Education. It’s going to utter waste, but that’s another story. To get it, I took Calc I-III, Linear Algebra, Statistics, Geometry, Abstract Algebra, and Discrete Math.

I’d like to pursue more on my own. I know I’ll have to refresh myself on Calc, but that’s cool because I kept all my textbooks, and that particular book is very good. So that’s no sweat.

I’d like to do some number theory. I took a correspondance course a few years ago in it, but the course didn’t go well, for a number of reasons, one of which was that I really didn’t like the text (I found it to be insufficient for self-learning, with too few examples and good explanations.) I wish I could remember which it is.

So I’m looking for a good Number Theory text, and recommendations on other subjects to look into and good texts for them. I should probably do some Diff Eq, even though I don’t particularly want to. I’m kind of intrigued by Group Theory and Set Theory, would like to look into more Symbolic Logic, and have heard about Real Analysis but don’t know much about it.

Any help is appreciated.

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I’m not sure if this is at all what you’re looking for at all, but my all time favorite book of all time is Godel, Escher, Bach, which has a lot of extremely interesting math.

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GEB’s a great book, but as a math textbook, it sucks.

Let’s see…

I’m not familiar with too many number theory texts, but I’d definitely consider getting one from Dover, seeing as very few of their books are over $20.

Boyce & DiPrima is widely considered the best introductory diffeq text, although that makes me think certain things about the quality of the others out there.

There are a handful of classic algebra texts, none of which are real great for self-study. I used Herstein, but without a professor and a class, I don’t know that I would’ve gotten very far in it. Ironic, that as much as I like algebra, I only know one book. Analysis seems to be more popular.

Real analysis is the branch of math that deals with everything you encountered in calculus: limits, derivatives, integrals, series, etc. But you prove things, instead of just using them. Rudin’s “Principles of Mathematical Analysis” is the classic introduction, and it’s not bad. He wrote a more advanced book, but I hear that it’s only intelligible to people who already know the subject matter.

Complex analysis is like real analysis, except you use complex numbers. It’s probably the strangest branch of math out there, and one of the hardest (IMO). No recommendations here, cause I’m still looking.

For logic and set theory, pick up some introductory book (Barwise & Etchmenedy would probably be a good choice), go through that, and then get Mendelson’s “Introduction to Mathematical Logic”. It’s not for beginners, but once you can read it, it’s good, and it’s full of information.

More recommendations on other branches as I think of them.

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Oh yeah, if you want to know everything out there, check out this site.

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There’s also excellent supplemental material to be found on the web – Drexel’s Math Forum ( http://mathforum.org/ ) is a great place to start. The internet library provides links on a variety of subjects. ‘Ask Dr. Math’ covers answers to questions that students have, though it leans more towards high school level.

I picked up a real analysis text for $1 at a college bookstore clearance sale and I’ve been studying it on my own for a while now. It’s not a great text, but I’ve used Interactive Real Analysis to help out at times (a link I found through the Drexel library).

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::sighs heavily as her nerd rating goes sky high::

My favorite algebra book is Gallian’s. It’s on about the same level as Herstein, but the result’s a lot more…hands on? There are more examples, and a lot of just neat stuff. It’s the one I go back to when other things have ceased to make sense.

Another option for real analysis is Wade. I’ve had profs refer to it as an “expanded Rudin”… which could be useful if you’re trying to go it on your own.

My best recommendation for finding text books is to work out where the nearest University math library is, and sit down and poke at what’s there. When you get down to the handful of classic texts, it’s really a matter of style which one you like best.

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Thanks for the pointers! I’ll check some of these out.

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I got Oystein Ore’s book on Number Theory from Dover Books – a good read (and how can you forget a name like Oystein Ore?)

I learned DiffEqs out of Boyce and diPrima, like ultrafilter. I don’t know if it’s the best, but it’s clear.

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What kind of net connection do you have? I have a rather large connection of maths notes (all collected from sources online which freely provide them and all with appropriate credit to the authors, so it’s completely legal) I could send you as a starter. I haven’t read through all of them, but what I’ve browsed is pretty good. I could send them to you (through various methods - ftp, msn transfers, etc.) but the problem is that they’re currently totalling about 140MB, which isn’t great for dial up connections. :slight_smile: (which I’ll be back on come tuesday, so this is a limited time offer). Alternately I could write up a list of what I have, send that to you, and you could let me know what you’re interested in.

I can also suggest some books:

Set theory and Symbolic logic: I use ‘Set Theory and Logic’, by Robert R. Stoll. Published by Dover books. It’s about $15, starts from basic set theory and moves all the way up through the construction of the real numbers, axiomatic set theory, etc. It covers a reasonable amount of symbolic logic, and gets as far as Goedel’s theorem and Turing machines (although it only covers the latter two briefly). I haven’t tried extensive learning from it - I mostly use it as a reference - but it seems well written and clear.

Group Theory: A Course in Group Theory, by John F Humphreys. Oxford Science Publications. I can’t reccomend this one enough if you want to learn group theory - it’s clearly written, precise, and covers a lot.

Real Analysis: My favourite book on the subject is Real Analysis, by N L Caruthers. It’s not absolutely stunning, but it’s a good solid textbook, with lots of information. I don’t however have that much experience with other real analysis textbooks (this isn’t through lack of interest - I just tend to browse random books rather than look at specific ones).

Differential Equations: What? You mean like… applied maths? [haughy pure mathematician tone] Proper mathematicians do not do applied maths.[/hpmt] :slight_smile:

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Oh. Forgot to mention - Real Analysis is a Cambridge University Press publication. So, if you’re in America, that and the group theory book may not be that easy to get a hold of.

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Hee! Well I’ll be honest, I’m not into applied math and don’t care about it…I’m totally on the pure math tip. So if not going into engineering means no Diff Eq, that’s great news for me! After all, there’s the old saying: “Learning differential equations is like having sex with a goat. You could do it, but why would you want to?”

Essentially, what I’m trying to do is prep myself for a masters in Math. Right now I can’t actually do it, but that situation may change soon. So I’m looking to brush up on the stuff I’ve already taken (I have all my old texts) and get a good start on some new stuff.